Many systems of interest are characterized by an input-to-response (ItR) function which can only be queried by expensive experiments or simulations. For example, the ship motion response induced by an input wave group is obtained from expensive model of computational fluid dynamics (CFD). If the ItR function is deterministic, the probability distribution of the response is then induced only by the (assumed) known probability distribution of the input. The quantification of extreme-event probability for such systems has been studied extensively in the context of Monte-Carlo simulations. Due to the expensive evaluations of ItR and rareness of the extreme events, many studies aim for reducing the number of required samples for the computation. While techniques such as importance sampling and control variate [1]} provide certain levels of acceleration, a more successful category of methods make use of the sequential Bayesian experimental design (BED) [2]} (or active learning [3]}), where the next-best samples are selected based on the existing information. Two key components of the sequential BED are a surrogate model (usually a Gaussian progress regression [4]}) to approximate the ItR, and an optimization problem maximizing a predefined acquisition function to select the next-best samples. Varying in details in the two components, a spectrum of sequential BED methods for the purpose of estimating extreme-event probability have been developed [5]}, [6]}, [7]}, [8]}, [9]}, [10]}.
